Fig. 1. Geometric interpretation for spikes in the group delay function of a signal at frequency locations on the unit circle close to a zero of its z-transform polynomial.

Fig. 2. Effects of zeros to spectra of a signal: (a) Hanning windowed real speech frame (phoneme /a/ in the word “party”), (b) zeros in polar coordinates, (c) magnitude spectrum and (d) group delay function. The zeros close to the unit circle are superimposed on spectrum plots to show the link between the zeros and the dips/spikes on the spectra.

Fig. 3. ZZT plots on the z-plane: (a) the time-domain signal, (b) ZZT plot in cartesian coordinates, (c) ZZT plot in polar coordinates, (d) ZZT plot in polar coordinates (right half-plane), (e) magnitude spectrum and (f) group delay function.

Fig. 4. ZZT and source–filter model of speech. Note that magnitude spectra added are in dBs.

Fig. 5. Effect of window location on ZZT and group delay function of a real speech signal: (a) time-domain signal and window locations, (b) corresponding ZZT representations and (c) corresponding group delay functions. Note that a Blackman window is used.

Fig. 6. GCI synchronous windowing: (a) time-domain waveform of the windowed signal, (b) corresponding ZZT representation and (c) amplitude spectrum and group delay scaled plotted together.

Fig. 7. Single excitation synthetic speech signal: (a) zeros on the z-plane in polar coordinates, (b) magnitude spectrum and group delay function.

Fig. 8. Effect of window shape to ZZT patterns and group delay functions. GCI synchronous windowing using several window shapes: rectangle, Hamming, Hanning, Blackman, Gaussian and Hanning–Poisson.

Fig. 9. GCI synchronous Hanning–Poisson asymmetric windowing. Each row includes the window shape used, the ZZT representation, the group delay function and the amplitude spectrum of the windowed signal.

Fig. 10. Effect of windowing size to group delay of a synthetic speech signal. Each (Blackman) window size is indicated on the window waveform on the top figure. The group delay function of the resulting windowed data for each window is presented with the window size indicated on the left-top corner of the figure.

Fig. 11. Group delay spectrogram: (a) Group delay and (b) magnitude spectrogram for the sentence “she has left for a great party today”.

Fig. 12. Remaining problems for group delay of GCI synchronously windowed data: (a) GCI synchronously windowed speech data, (b) magnitude spectrum, (c) chirp group delay function. ZZT representation is presented on the right column both in cartesian and polar coordinates.

Fig. 13. GCI-synchronous chirp group delay function estimation: (a) ZZT after suppression of ZZT outside the unit circle and (b) chirp group delay.

Fig. 14. GCI-asynchronous chirp group delay function estimation: (a) ZZT after suppression of ZZT outside the unit circle and (b) chirp group delay (CGDZP).

Fig. 15. Time-domain signal of a 30 ms speech frame and its group delay function. The frame example is extracted from the noise-free utterance “mah_4625” of the test set A of the AURORA-2 (Hirsch and Pearce, 2000) and corresponds to vowel/i/ in word “six”.

Fig. 16. Power spectrum (PowerS) and group delay representations for the speech signal frame in Fig. 15.

Fig. 17. Spectrogram plots of a noise-free utterance. Only the first half of the file, “mah_4625a”, that contains the digit utterance “4 6” is presented.

Methods for ASR feature extraction based on power spectrum or group delay function

All methods are applied on frames of length 30 ms and shifted by 10 ms.

ASR performances for various feature extractions on the AURORA-2 task

Results are given in terms of word error rate (WER) in percent.

ASR performances for features combined with MFCC on the AURORA-2 task

Results are given in terms of word error rate (WER) in percent.